On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.     

Capillarity driven spreading of power-law fluids

We investigate the spreading of thin liquid films of power-law rheology. We construct an explicit travelling wave solution and source-type similarity solutions. We show that when the nonlinearity exponent λ for the rheology is larger than one, the governing dimensionless equation h_t + (h^λ+2|h_xxx|^λ−1h_xxx)_x=0 admits solutions with compact support and moving fronts. We also show that the solutions have bounded energy dissipation rate.     

On power-law fluids with the power-law index proportional to the pressure

In this short note we study special unsteady flows of a fluid whose viscosity depends on both the pressure and the shear rate. Here we consider an interesting dependence of the viscosity on the pressure and the shear rate; a power-law of the shear rate wherein the exponent depends on the pressure. The problem is important from the perspective of fluid dynamics in that we obtain solutions to a technologically relevant problem, and also from the point of view of mathematics as the analysis of the problem rests on the theory of spaces with variable exponents. We use the theory to prove the existence of solutions to generalizations of Stokes’ first and second problem.     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

Lubrication approximation of flows of a special class of non-Newtonian fluids defined by rate type constitutive equations

We consider the flow of a rate-type fluid defined by an implicit constitutive equation in a channel with non flat walls. We also assume that the channel characteristic width is small in comparison to the channel length, so that the lubrication approximation can be applied. The model developed is mainly motivated by the evidence that many lubricants seem to be well described by implicit rate type fluid models. The mathematical problem is reduced to an integro-differential equation for the pressure that is solved numerically for several channel profiles.     

Concave solutions of a general self-similar boundary layer problem for power-law fluids

In this paper, we give a rigorous mathematical analysis for a third order nonlinear boundary value problem. The boundary value problem can be applied to steady free convection around a vertical impermeable flat plate in a fluid-saturated porous medium, or steady flow of a power-law fluid induced by impermeable stretching walls in the frame of boundary layer approximation. We establish the uniqueness, existence and nonexistence of (normal) concave solutions or generalized concave solutions to the problem, and obtain some results about boundedness and asymptotic behavior of the (normal) concave solution or the generalized concave solution.     

Boundary layer flow of power-law fluids past arbitrary profiles

Two-dimensional steady-state boundary layer equations of power-lawfluids are derived using a special coordinate system which makesthe equations independent of the body shape immersed in theflow. In deriving the boundary layer equations, the method ofmatched asymptotic expansions is used. It is shown that thesimilarity solutions for power-law fluids are much the sameas those of Newtonian fluids. Similarity solutions correspondingto the case of parallel flow past a flat plate and stagnation-pointflow are presented. Finally, the shear stress is calculatedfor different geometries.     

On a class of special flows

Summary A special flow, in which the basic automorphism is a Bernoulli automorphism and the ceiling function depends only on the present position of the Bernoulli sequence and is not lattice distributed, is a K-flow.This paper represents the result, in a revised form, which was presented at the Symposium on Ergodic Theory held at Mathematisches Forschungsinstitut Oberwolfach from August 4 to 10, 1968.     

Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0⁺ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J₀(•), Y₀(•), J₁(•), Y₁(•), J₂(•) and Y₂(•). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.     

Velocity fields with power-law spectra for modeling turbulent flows

We consider a generalization of homogeneous and isotropic Çinlar velocity fields to capture power-law spectra. The random velocity field is non-Gaussian with a representation motivated by Lagrangian and Eulerian observations. A wide range of turbulent flows can be generated by varying the stochastic parameters of the model. The velocity field being a functional version of Poisson shot-noise is constructed as the superposition of eddies randomized through their types and arrival times. We introduce a dependence between the eddy types which are spatial parameters and the decay parameter which is temporal. As a result, long-range correlation in space and a power-law spectrum previously used with Ornstein–Uhlenbeck velocity fields are achieved. We show that a corresponding power-law form for the probability distribution of the eddy diameter is sufficient for this result. The parameters of the probability distribution are further specified in view of Kolmogorov theory of the inertial scales. In particular, ∣k∣^−5/3 scaling of the spectrum is obtained. In the diffusive limit, we show that the parameters governing the decay and the arrival rate, and the speed of rotation of an eddy increase while its diameter decreases. That is, the eddies arrive fast, decay fast, and rotate fast with a small radius for a Brownian limit.     

Similarity solutions for free convection to power-law fluids from a heated vertical plate

Steady, laminar free-convection boundary-layer flow of a power-law fluid over a vertical heated plate under mixed thermal boundary conditions is investigated. A transformation relating the similarity solutions of the boundary-layer velocity and temperature profiles is obtained.     

Generalized Crane flows of micropolar fluids

The hydromagnetic flow induced by a continuous stretching surface in a quiescent micropolar fluid is revisited in this Note. It is shown that the problem admits an exact analytical solution for arbitrary differentiable stretching velocities when the surface is permeable and a suitable lateral suction/injection of the fluid is applied. This result generalizes the classical Crane-type solutions which hold for linear stretching velocities only.     

Unsteady helical flows of Oldroyd-B fluids

The unsteady helical flow of an Oldroyd-B fluid, in an infinite circular cylinder, is studied by using finite Hankel transforms. The motion is produced by the cylinder that, at time t = 0⁺, is subject to torsional and longitudinal time-dependent shear stresses. The solutions that have been obtained, presented under series form, satisfy all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion is underlined by graphical illustrations.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.Received: December 16, 2003; revised: October 11, 2004     

Shear instability for stratified hydrostatic flows

Stratified flows in hydrostatic balance are studied in both their multilayer and continuous formulations. A novel stability criterion is proposed for stratified flows, which reinterprets stability in terms not of growth of small perturbations but of the well‐posedness of the time evolution. This reinterpretation allows one to extend the classic results of Miles and Howard concerning steady and planar flows to the realm of flows that are nonuniform and unsteady. © 2008 Wiley Periodicals, Inc.